Find all zeros of the function $f ( x ) = ( x + 6 ) ( x + 3 i ) ( x - 3 i )$.
A) $x = 6 , - 3 i , 3 i$
B) $x = - 6,3 i$
C) $x = - 6 , - 3,3$
D) $x = - 6 , - 3 i , 3 i$
E) $x = - 6$

Question

Quizplus · Accepted Answer

To find the zeros of the function, we set $f(x)$ equal to zero and solve for $x$. \begin{align*}
f(x) &= (x + 6)(x + 3i)(x - 3i)\
&= (x + 6)(x^2 + 9)\
&= x^3 + 6x^2 + 9x + 54
\end{align*} Setting this equal to zero, we get \begin{align*}
x^3 + 6x^2 + 9x + 54 &= 0\
x^3 + 6x^2 + 9x &= -54\
x(x^2 + 6x + 9) &= -54\
x(x + 3)^2 &= -54
\end{align*} We see that $x$ must be negative since the left-hand side of the equation is always nonnegative while the right-hand side is negative. Therefore, the only zero(s) of the function is/are $x = -6, -3i, 3i$. So, the answer is $\boxed{	extbf{(D)}}$.

Find All Zeros of the Function f(x)=(x+6)(x+3i)(x−3i)f ( x ) = ( x + 6 ) ( x + 3 i ) ( x - 3 i )f(x)=(x+6)(x+3i)(x−3i)

Find All Zeros of the Function $f ( x ) = ( x + 6 ) ( x + 3 i ) ( x - 3 i )$